This paper proposes an elegant optimization framework consisting of a mix of linear-matrix-inequality and second-order-cone constraints. The proposed framework generalizes the semidefinite relaxation (SDR) enabled solution to the typical transmit beamforming problems presented in the form of quadratically constrained quadratic programs (QCQPs) in the literature. It is proved that the optimization problems subsumed under the framework always admit a rank-one optimal solution when they are feasible and their optimal solutions are not trivial. This finding indicates that the relaxation is tight as the optimal solution of the original beamforming QCQP can be straightforwardly obtained from that of the SDR counterpart without any loss of optimality. Four representative examples of transmit beamforming, i.e., transmit beamforming with perfect channel state information (CSI), transmit beamforming with imperfect CSI, chance-constraint approach for imperfect CSI, and reconfigurable-intelligent-surface (RIS) aided beamforming, are shown to demonstrate how the proposed optimization framework can be realized in deriving the SDR counterparts for different beamforming designs.

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